A simplest possible model involving only tumor cells and killer lymphocytes was developed and extensively studied. It was found that even in this simple two-variable system, the behavior is robust and includes the possibility of tumor-host coexistence at sufficiently low tumor levels, and oscillatory behavior. The predicted results of immune suppression or stimulation are in general agreement with observation and intuitive expectations. In addition, however, the theory predicts that there will be isolated instances under which an increase in lymphocyte concentration may assist tumor growth. This phenomonon is also observed experimentally and the theory provides a possible explanation (and so far as I know, the only one) which is logically consistent with immune surveillance. The model in its most primitive form, however, does not allow de novo development. It was shown that making the model slightly more realistic by introducing a population of precursor lymphocytes which must go through some maturation period before they are effective in destroying tumor cells, removes this difficulty.